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Enumeration of binary trees and universal types
 Discrete Math. Theor. Comput. Sci
"... Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., heig ..."
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Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a given path length (sum of depths) are there? This question arose in the study of universal types of sequences. Two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the LempelZiv’78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, Tp, of given path length p (and also the number of distinct LempelZiv’78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to Tp ∼ 2 2p/(log 2 p)(1+O(log−2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrases in the LempelZiv’78 scheme) when a tree is selected randomly among all trees of given path length p. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well
On the joint path length distribution in random binary trees
 Stud. Appl. Math
"... During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right p ..."
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During the 10th Seminar on Analysis of Algorithms, MSRI, Berkeley, June 2004, Knuth posed the problem of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent. This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n 5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution’s tails and moments. We shall also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method.
Enumeration of Binary Trees and Universal Types*
, 2004
"... Abstract Binary unlabeled ordered trees (further called binary trees) were studied at leastsince Euler, who enumerated them. The number of such trees with n nodes is nowknown as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g ..."
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Abstract Binary unlabeled ordered trees (further called binary trees) were studied at leastsince Euler, who enumerated them. The number of such trees with n nodes is nowknown as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions fora randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated byinformation theory considerations: how many binary trees of a given path length (sum of depths) are there? This question arose in the study of universal types of sequences.Two sequences of length p have the same universal type if they generate the same setof phrases in the incremental parsing of the LempelZiv'78 scheme since one proves that such sequences converge to the same empirical distribution. It turns out that the numberof distinct types of sequences of length p corresponds to the number of binary (unlabeledand ordered) trees, Tp, of given path length p (and also the number of distinct LempelZiv'78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to Tp, 22p/(log2 p)(1+O(log2/3 p)). Then we establish various limiting distributions for the number of nodes (number of phrasesin the LempelZiv'78 scheme) when a tree is selected randomly among all trees of given
Enumeration of Binary Trees, LempelZiv’78 Parsings, and Universal Types ∗
"... Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., heig ..."
Abstract
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Binary unlabeled ordered trees (further called binary trees) were studied at least since Euler, who enumerated them. The number of such trees with n nodes is now known as the Catalan number. Over the years various interesting questions about the statistics of such trees were investigated (e.g., height and path length distributions for a randomly selected tree). Binary trees find an abundance of applications in computer science. However, recently Seroussi posed a new and interesting problem motivated by information theory considerations: how many binary trees of a given path length (sum of depths) are there? This question arose in the study of universal types of sequences. Seroussi declares that two sequences of length p have the same universal type if they generate the same set of phrases in the incremental parsing of the LempelZiv’78 scheme. (He then proves that sequences of the same type converge to the same empirical distribution.) It turns out that the number of distinct types of sequences of length p corresponds to the number of binary (unlabeled and ordered) trees, Tp, of given path length p (and also the number of different LempelZiv’78 parsings of length p sequences). We first show that the number of binary trees with given path length p is asymptotically equal to Tp ∼ 2 2p/(log 2 p). Then we establish various limiting distributions for the number of nodes (number of phrases in the LempelZiv’78 scheme) when a tree is selected randomly among all Tp trees. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method and matched asymptotics. 1
Binary Trees, Left and Right Paths, WKB Expansions, and Painlevé
"... of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the di ..."
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of analyzing the left and the right path length in a random binary trees. In particular, Knuth asked about properties of the generating function of the joint distribution of the left and the right path lengths. In this paper, we mostly focus on the asymptotic properties of the distribution of the difference between the left and the right path lengths. Among other things, we show that the Laplace transform of the appropriately normalized moment generating function of the path difference satisfies the first Painlevé transcendent. This is a nonlinear differential equation that has appeared in many modern applications, from nonlinear waves to random matrices. Surprisingly, we find out that the difference between path lengths is of the order n5/4 where n is the number of nodes in the binary tree. This was also recently observed by Marckert and Janson. We present precise asymptotics of the distribution’s tails and moments. We shall also discuss the joint distribution of the left and right path lengths. Throughout, we use methods of analytic algorithmics such as generating functions and complex asymptotics, as well as methods of applied mathematics such as the WKB method. 1
Analysis of Algorithms (AofA): Part I: 1993  1998 ("Dagstuhl Period")
"... This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on ..."
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This is the rst installment of the Algorithmics Column dedicated to Analysis of Algorithms (AofA) that sometimes goes under the name AverageCase Analysis of Algorithms or Mathematical Analysis of Algorithms. The area of analysis of algorithms (at least, the way we understand it here) was born on July 27, 1963, when D. E. Knuth wrote his \Notes on Open Addressing". Since 1963 the eld has been undergoing substantial changes. We report here how it evolved since then. For a long time this area of research did not have a real \home". But in 1993 the rst seminar entirely devoted to analysis of algorithms took place in Dagstuhl, Germany. Since then seven seminars were organized, and in this column we briey summarize the rst three meetings held in Schloss Dagstuhl (thus \Dagstuhl Period") and discuss various scienti c activities that took place, describing some research problems, solutions, and open problems discussed during these meetings. In addition, we describe three special issues dedicated to these meetings.
HACHAGE, ARBRES, CHEMINS & GRAPHES
"... Mathématiques discrètes et continues se rencontrent et se complètent volontiers harmonieusement. C’est cette thèse que nous voudrions illustrer en discutant un problème classique aux ramifications nombreuses—l’analyse du hachage avec essais linéaires. L’exemple est issu de l’analyse d’algorithmes, d ..."
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Mathématiques discrètes et continues se rencontrent et se complètent volontiers harmonieusement. C’est cette thèse que nous voudrions illustrer en discutant un problème classique aux ramifications nombreuses—l’analyse du hachage avec essais linéaires. L’exemple est issu de l’analyse d’algorithmes, domaine fondé par Knuth et qui se situe luimême “à cheval ” entre l’informatique, l’analyse combinatoire, et la théorie des probabilités. Lors de son traitement se croisent au fil du temps des approches très diverses, et l’on rencontrera des questions posées par Ramanujan à Hardy en 1913, un travail d’été de Knuth datant de 1962 et qui est à l’origine de l’analyse d’algorithmes en informatique, des recherches en analyse combinatoire du statisticien Kreweras, diverses rencontres avec les modèles de graphes aléatoires au sens d’Erdös et Rényi, un peu d’analyse complexe et d’analyse asymptotique, des arbres qu’on peut voir comme issus de processus de GaltonWatson particuliers, et, pour finir, un peu de processus, dont l’ineffable mouvement Brownien! Tout ceci contribuant in fine à une compréhension très précise d’un modèle simple d’aléa discret.